The twist for positroid varieties

Transcription

1 The twist for positroid varieties Greg Muller, joint with David Speyer July 7, 206

2 2-colored graphs in the disc Let G be a graph in the disc with... a bipartite 2-coloring of its internal vertices, and a clockwise indexing of its boundary vertices from to n Assume (for simplicity) Each boundary vertex is adjacent to one vertex, which is white.

3 Matchings of G For this talk, a matching of G is a subset of the edges for which every internal vertex is in exactly one edge Easy observation Every matching of G contains k := white vertices black vertices boundary vertices.

4 Two maps, both alike in dignity In an unpublished note in 2003, Postnikov associated two maps to such a graph in the disc, which may be defined using matchings. A boundary measurement map B : an algebraic torus a positroid variety A cluster: a rational map F : a positroid variety an algebraic torus These maps have proven fruitful in studying flows in networks, total positivity, elementary factorizations of matrices, integrable systems and perturbative field theories. Despite abundant applications, basic questions about these maps remained open for more than a decade; like, how are they related?

5 Weighted enumeration of matchings Question What are the matchings of G with boundary I {, 2,..., n}? We encode the answer in the partition function Z I of I. 2 Z I : (C ) Edges(G) C an edge weighting w i 3 4 d a b e f g c h total weight of matchings with boundary I n o j k l Z m 356 (w) = 5 s p q t r u. 6

6 Weighted enumeration of matchings Question What are the matchings of G with boundary I {, 2,..., n}? We encode the answer in the partition function Z I of I. Z I : (C ) Edges(G) C an edge weighting w 3 4 a total weight of matchings with boundary I adfprou d p f r h o 5 Z 356 (w) = u. 6

7 Weighted enumeration of matchings Question What are the matchings of G with boundary I {, 2,..., n}? We encode the answer in the partition function Z I of I. 2 Z I : (C ) Edges(G) C an edge weighting w 3 4 a d k q l u h o 5 total weight of matchings with boundary I Z 356 (w) = adfprou + adhklqou

8 Weighted enumeration of matchings Question What are the matchings of G with boundary I {, 2,..., n}? We encode the answer in the partition function Z I of I. 2 Z I : (C ) Edges(G) C an edge weighting w 3 4 a j e q l u h o 5 total weight of matchings with boundary I adfprou + adhklqou + Z 356 (w) = aehjlqou

9 Weighted enumeration of matchings Question What are the matchings of G with boundary I {, 2,..., n}? We encode the answer in the partition function Z I of I. 2 Z I : (C ) Edges(G) C an edge weighting w 3 4 a d k q g u m o 5 total weight of matchings with boundary I adfprou + adhklqou + Z 356 (w) = aehjlqou + adgkmqou

10 Weighted enumeration of matchings Question What are the matchings of G with boundary I {, 2,..., n}? We encode the answer in the partition function Z I of I.. 2 Z I : (C ) Edges(G) C an edge weighting w 3 4 a j e q g u m o 5 total weight of matchings with boundary I adfprou + adhklqou + Z 356 (w) = aehjlqou + adgkmqou + aegjmqou 6

11 The Plücker relations The partition functions satisfy a surprising family of identities. Theorem (Kuo 04, Postnikov 06) The partition functions {Z I } satisfy the Plücker relations. Concretely, this means a k n matrix A w such that I, Z I (w) = I (A w ) := determinant of columns of A w in I A w = 0 aep fmop klqu+fpru bks 0 klns klst adk+aej 0 bik 0 fjmo ikln fjru iklt bciklnst bikost(hl+gm) bgiknrsu

12 The Plücker relations The partition functions satisfy a surprising family of identities. Theorem (Kuo 04, Postnikov 06) The partition functions {Z I } satisfy the Plücker relations. Concretely, this means a k n matrix A w such that I, Z I (w) = I (A w ) := determinant of columns of A w in I A w = 0 aep fmop klqu+fpru bks 0 klns klst adk+aej 0 bik 0 fjmo ikln fjru iklt bciklnst bikost(hl+gm) bgiknrsu

13 The Plücker relations The partition functions satisfy a surprising family of identities. Theorem (Kuo 04, Postnikov 06) The partition functions {Z I } satisfy the Plücker relations. Concretely, this means a k n matrix A w such that I, Z I (w) = I (A w ) := determinant of columns of A w in I 0 aep fmop klqu+fpru bks 0 klns klst A w = adk+aej 0 bik 0 fjmo ikln fjru iklt bciklnst bikost(hl+gm) bgiknrsu 356 (A w ) = adfprou + adhklqou + aehjlqou + adgkmqou + aegjmqou = Z I (w)

14 The boundary measurement map While A w is not uniquely determined, the span of its rows is! The Z I (w) are encoded (up to scaling) in the well-defined subspace B(w) := rowspan(a w ) C n We get a map to Gr(k, n), the Grassmannian of k-spaces in C n. The boundary measurement map (raw version) B : (C ) Edges(G) Gr(k, n) an edge weighting w rowspan(a w )

15 Restricting the target The image of B is typically much smaller than Gr(k, n), for the simple reason that some of the Z I can be zero. The positroid variety of G Let Π(G) Gr(k, n) parametrize rowspans of matrices A with I (A) = 0 whenever G has no matchings with boundary I Every I except 23 is the boundary of at least one matching in G on the left. 0 0 Π(G) = rowspan

16 Quotienting the domain B also factors through a simple symmetry of the edge weights. Gauge transformations A gauge transformation at an internal vertex v multiplies the weight of each edge adjacent to v by some fixed scalar λ C. These transformations do not change the subspace B(w). The boundary measurement map (refined version) B : (C ) Edges(G) /Gauge Π(G)

17 Reduced graphs The map B is most interesting for reduced graphs. Reduced graphs The graph G is reduced if every face is a disc, and there is no G with fewer faces and Π(G) = Π(G ). Intuitively, these are the graphs which realize Π(G) most efficiently. Expectation When G is reduced, the map is an open inclusion of varieties. B : (C ) Edges(G) /Gauge Π(G) Applications are particularly interested in a constructive answer: given [V ] in Π(G), construct w such that B(w) = V.

18 To attack this problem, explore the geometry of Π(G) and construct our second map of interest, we ask... Question How can we describe a point in Π(G)?

19 Coordinates on Π(G) Plücker coordinates Given a matrix A, rowspan(a) is determined by the Plücker coordinates { I (A)}, running over all I {, 2,..., n} of size k. However, there is a large amount of redundancy in these numbers. Observation If G is reduced, the dimension of Π(G) is Faces(G). So, at least Faces(G) -many Plücker coordinates are needed to determine a point in Π(G). Question Can we determine a point in Π(G) with exactly Faces(G) -many Plücker coordinates?

20 Matchings associated to faces Idea: use boundaries of distinguished matchings to choose the I. Lemma (M-Speyer) Given reduced G and a face f, there! matching M f containing every edge in f going clockwise from white to black, and 2 one fewer than half the edges around every other face f 5 6

21 Matchings associated to faces Idea: use boundaries of distinguished matchings to choose the I. Lemma (M-Speyer) Given reduced G and a face f, there! matching M f containing every edge in f going clockwise from white to black, and 2 one fewer than half the edges around every other face f 5 6

22 Matchings associated to faces Idea: use boundaries of distinguished matchings to choose the I. Lemma (M-Speyer) Given reduced G and a face f, there! matching M f containing every edge in f going clockwise from white to black, and 2 one fewer than half the edges around every other face f 5 6

23 Matchings associated to faces Idea: use boundaries of distinguished matchings to choose the I. Lemma (M-Speyer) Given reduced G and a face f, there! matching M f containing every edge in f going clockwise from white to black, and 2 one fewer than half the edges around every other face f 5 6

24 Matchings associated to faces Idea: use boundaries of distinguished matchings to choose the I. Lemma (M-Speyer) Given reduced G and a face f, there! matching M f containing every edge in f going clockwise from white to black, and 2 one fewer than half the edges around every other face f 5 6

25 Plücker coordinates associated to faces Each face determines a Plücker coordinate. f := Mf We combine these into a map on Π(G). F : Π(G) C Faces(G) /Scaling Expectation F restricts to an isomorphism on the subset where no f is zero. F : Π(G) (C ) Faces(G) /Scaling This is a consequence of a deeper conjecture (which we ll ignore). Conjecture [Scott, Postnikov, M-Speyer, LeClerc] The homogeneous coordinate ring of Π(G) is a cluster algebra, and the Plücker coordinates of the faces of G form a cluster

26 Two conjectural tori associated to a reduced graph So, a reduced graph G determines two maps, and each map conjecturally defines an open subset in Π(G). The image of the boundary measurement map B : (C ) Edges(G) /Gauge Π(G) The domain of definition of the cluster of Plücker coordinates Natural question F : Π(G) (C ) Faces(G) /Scaling What is the relation between these two subsets? In a simple world, they d coincide and we d have an isomorphism F B : (C ) Edges(G) /Gauge (C ) Faces(G) /Scaling

27 The need for a twist In the real world, we need a twist automorphism τ of Π(G), which will fit into a composite isomorphism (C ) Edges(G) /Gauge (C ) Faces(G) /Scaling B Π(G) τ Π(G) F and thus take the image of B to the domain of F.

28 The twist of a matrix Let A be a k n matrix of rank k, and assume no zero columns. Denote the ith column of A by A i, with cyclic indices: A i+n = A i. Definition: The twist The twist τ(a) of A is the k n-matrix defined on columns by τ(a) i A i = τ(a) i A j = 0, if A j is not in the span of {A i, A i+,..., A j } 0 0???? 0 2???? 0 0 0????

29 The twist of a matrix Let A be a k n matrix of rank k, and assume no zero columns. Denote the ith column of A by A i, with cyclic indices: A i+n = A i. Definition: The twist The twist τ(a) of A is the k n-matrix defined on columns by τ(a) i A i = τ(a) i A j = 0, if A j is not in the span of {A i, A i+,..., A j } 0 0??? 0 2???? 0 0 0????

30 The twist of a matrix Let A be a k n matrix of rank k, and assume no zero columns. Denote the ith column of A by A i, with cyclic indices: A i+n = A i. Definition: The twist The twist τ(a) of A is the k n-matrix defined on columns by τ(a) i A i = τ(a) i A j = 0, if A j is not in the span of {A i, A i+,..., A j } 0 0??? 0 2??? 0 0 0????

31 The twist of a matrix Let A be a k n matrix of rank k, and assume no zero columns. Denote the ith column of A by A i, with cyclic indices: A i+n = A i. Definition: The twist The twist τ(a) of A is the k n-matrix defined on columns by τ(a) i A i = τ(a) i A j = 0, if A j is not in the span of {A i, A i+,..., A j } 0 0??? 0 2??? 0 0 0????

32 The twist of a matrix Let A be a k n matrix of rank k, and assume no zero columns. Denote the ith column of A by A i, with cyclic indices: A i+n = A i. Definition: The twist The twist τ(a) of A is the k n-matrix defined on columns by τ(a) i A i = τ(a) i A j = 0, if A j is not in the span of {A i, A i+,..., A j } 0 0??? 0 2??? ???

33 The twist of a matrix Let A be a k n matrix of rank k, and assume no zero columns. Denote the ith column of A by A i, with cyclic indices: A i+n = A i. Definition: The twist The twist τ(a) of A is the k n-matrix defined on columns by τ(a) i A i = τ(a) i A j = 0, if A j is not in the span of {A i, A i+,..., A j }

34 Properties of the twist Lemma (M-Speyer) The twist has an inverse given by the same formula as τ, except moving to the left instead of the right. Theorem (M-Speyer) The twist map descends to a piecewise-regular automorphism τ : Π(G) Π(G)

35 The twist makes everything work! Theorem (M-Speyer) If G is reduced, there is a combinatorially-defined isomorphism such that the following diagram commutes. (C ) Edges(G) /Gauge (C ) Faces(G) /Scaling B Π(G) τ F Π(G) Corollaries! The boundary measurement map B is an open inclusion. The cluster F is an isomorphism on its domain.

36 Application: Inverting the boundary measurement map Let s invert B in a classic example! Example: The unipotent cell in GL(3), as a positroid cell?? 3 4????? 2 5???????? 6 B 0 0 a b c d e f

37 Application: Inverting the boundary measurement map Let s invert B in a classic example! Example: The unipotent cell in GL(3), as a positroid cell 0 0 a b c d e f

38 Application: Inverting the boundary measurement map Let s invert B in a classic example! Example: The unipotent cell in GL(3), as a positroid cell 0 0 a b c d e f τ 0 0 e a bd ce c b b c 0 0 d ad be cd e be cd be cd f 0 0 df adf

39 Application: Inverting the boundary measurement map Let s invert B in a classic example! Example: The unipotent cell in GL(3), as a positroid cell 3 4 c a 2 be cd b acd (adf ) ad a b c d e f τ 0 0 e a bd ce c b b c 0 0 d ad be cd e be cd be cd f 0 0 df adf F

40 Application: Inverting the boundary measurement map Let s invert B in a classic example! Example: The unipotent cell in GL(3), as a positroid cell 3 a 4 ac 2 c ac 2 d a 2 cd b b ad 5 acd(be cd) b a 2 cd 2 f b be cd adf (be cd) a 2 cd 2 f b a 2 d 2 f a 2 cd 2 b adf c a 2 be cd b acd (adf ) ad a b c d e f τ 0 0 e a bd ce c b b c 0 0 d ad be cd e be cd be cd f 0 0 df adf F

41 Application: Inverting the boundary measurement map Let s invert B in a classic example! Example: The unipotent cell in GL(3), as a positroid cell a b d c a d be cd cd b bf bf be cd acd ad f 6 6 (adf ) B 0 0 a b c d e f τ 0 0 e a bd ce c b b c 0 0 d ad be cd e be cd be cd f 0 0 df adf F

42 Relation to the Chamber Ansatz We have the following boundary measurement map computation. 3 a 4 b d 2 d 5 be cd cd bf bf f 6 B 0 0 a b c d e f This is equivalent to a factorization into elementary matrices. [ ] [ ] [ ] [ ] [ ] a b c 0 0 b 0 d 0 0 a 0 0 be cd cd 0 d e = d 0 bf bf 0 0 f f